We consider the following class of three point boundary value problem

In recent years, multipoint boundary value problems have been extensively studied by many authors ([

It is well known that one of the most important tools for dealing with existence results for nonlinear problems is the method of upper and lower solutions. The method of upper and lower solutions has a long history and some of its ideas can be traced back to Picard [

Recently, there have been numerous results in the presence of an upper solution

Recently, Li et al. [

The present work proves some new existing results for three point BVPs. Our technique is based on Picard-type iterative scheme and is quite simple and efficient from computational point of view. We believe that it can be very well adapted for this type of problem. In this paper we consider the following three point BVP:

The paper is divided into 4 sections. In Section

To investigate (

Let us assume that

It is easy to see that

Green's function for the following linear three point BVP

See proof of Lemma 2.1 in [

When

See proof of Lemma 2.3 in [

Particulary

Assume that

It is easy to see that

Green's function for the following three point BVP

Proof is same as given in Lemma

When

Proof is same as given in Lemma

Let

Let

Based on maximum and antimaximum Principle we develop theory to solve the three point nonlinear BVP and divide it into the following two subsections.

Let there exist

From (

In view of

Now from (

Now assuming that

From (

Using Lemma

Any solution

Let there exist

Proof follows from the analysis of Theorem

To verify our results, we consider examples and show that there exists at least one value of

Consider the boundary value problem

Plot of

Plot of

Plot of

Plot of

Consider the boundary value problem

Plot of

Plot of

Plot of

Plot of

The monotone iterative technique coupled with upper and lower solutions is a powerful tool for computation of solutions of nonlinear three point boundary value problems. It proves the existence of solutions analytically and gives us a tool so that numerical solutions can also be computed and then some real-life problems, for example, bridge design problem, thermostat problem, and so forth, can be solved. We have plotted sequences for both

This work is partially supported by Grant provided by UGC, New Delhi, India, File no. F.4-1/2006 (BSR)/7-203/2009(BSR).